Consider the theory ${\rm PA}^{\mathbb{T}}$ obtained by adding a truth predicate to Peano arithmetic, applicable to sentences of the unaugmented language and satisfying the compositionality axioms $\mathbb{T}(A \wedge B) \leftrightarrow (\mathbb{T}(A) \wedge \mathbb{T}(B))$, $(\forall n)\mathbb{T}(A(n)) \leftrightarrow \mathbb{T}((\forall n)A(n))$, etc., together with the induction scheme for all formulas of the augmented language.

I've read that this theory proves the same arithmetical formulas as ${\rm ACA}$, which I guess implies that it proves induction up to anything less than $\varepsilon_{\varepsilon_0}$ for all arithmetical formulas. So $\varepsilon_{\varepsilon_0}$ is the proof-theoretic ordinal of ${\rm PA}^{\mathbb{T}}$ in this sense.

Suppose we iterate this construction: let ${\rm PA}^{\mathbb{T}^2}$ be obtained by adding a new truth predicate, applicable to sentences of ${\rm PA}^{\mathbb{T}}$, with compositionality axioms and the induction scheme for all formulas of the newly augmented language.

What is its proof-theoretic ordinal (in the sense of: proving induction up to what for all arithmetical formulas)? What if we iterate the construction finitely many times?

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